Proportional electroluminescence in noble gases is used in two-phase detectors for dark-matter searches for detecting ionization signals in a liquid phase (S2 signals). Time features of electroluminescence signals in a two-phase argon detector were studied systematically both over a broad range of reduced electric fields—from 3 to 9 Td (Townsend)—and for various methods of optical readout in different spectral ranges—by means of cryogenic photomultiplier tubes (PMTs) and silicon photomultipliers (SiPMs), with a wavelength shifter and without it, in vacuum ultraviolet and visible ranges. Both a fast and a slow component of signals were observed. An observation of an unusual long component of the S2 signal is described in the present article. This component has a time constant of about 40 μs, which grows with increasing electric field, and can not be explained in terms of known scintillation mechanisms. Also, it is shown here that the fast component can be used to measure the thickness of the electroluminescence gap to a submillimeter precision. The results obtained in these experiments may find practical applications in the DarkSide experiment devoted to searches for dark matter.